Abstract
It has been conjectured that every Steiner triple system of order v 6= 7 has chromatic index at most (v + 3)=2 when v = 3 (mod 6) and at most (v + 5)=2 when v = 1 (mod 6). Herein, we construct a Steiner triple system of order v with chromatic index at least (v + 3)=2 for each integer v = 3 (mod 6) such that v ≥ 15, with four possible exceptions. We further show that the maximum number of disjoint parallel classes in the systems constructed is sublinear in v. Finally, we establish for each order v ≥ 15 (mod 18) that there are at least vv2(1=6+o(1)) nonisomorphic Steiner triple systems with chromatic index at least (v + 3)=2 and that some of these systems are cyclic.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 2603-2611 |
| Number of pages | 9 |
| Journal | SIAM Journal on Discrete Mathematics |
| Volume | 31 |
| Issue number | 4 |
| DOIs | |
| State | Published - Jan 1 2017 |
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Keywords
- Block coloring
- Chromatic index
- Parallel class
- Steiner triple system
ASJC Scopus subject areas
- Mathematics(all)
Cite this
Steiner triple systems with high chromatic index. / Bryant, Darryn; Colbourn, Charles; Horsley, Daniel; Wanless, Ian M.
In: SIAM Journal on Discrete Mathematics, Vol. 31, No. 4, 01.01.2017, p. 2603-2611.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Steiner triple systems with high chromatic index
AU - Bryant, Darryn
AU - Colbourn, Charles
AU - Horsley, Daniel
AU - Wanless, Ian M.
PY - 2017/1/1
Y1 - 2017/1/1
N2 - It has been conjectured that every Steiner triple system of order v 6= 7 has chromatic index at most (v + 3)=2 when v = 3 (mod 6) and at most (v + 5)=2 when v = 1 (mod 6). Herein, we construct a Steiner triple system of order v with chromatic index at least (v + 3)=2 for each integer v = 3 (mod 6) such that v ≥ 15, with four possible exceptions. We further show that the maximum number of disjoint parallel classes in the systems constructed is sublinear in v. Finally, we establish for each order v ≥ 15 (mod 18) that there are at least vv2(1=6+o(1)) nonisomorphic Steiner triple systems with chromatic index at least (v + 3)=2 and that some of these systems are cyclic.
AB - It has been conjectured that every Steiner triple system of order v 6= 7 has chromatic index at most (v + 3)=2 when v = 3 (mod 6) and at most (v + 5)=2 when v = 1 (mod 6). Herein, we construct a Steiner triple system of order v with chromatic index at least (v + 3)=2 for each integer v = 3 (mod 6) such that v ≥ 15, with four possible exceptions. We further show that the maximum number of disjoint parallel classes in the systems constructed is sublinear in v. Finally, we establish for each order v ≥ 15 (mod 18) that there are at least vv2(1=6+o(1)) nonisomorphic Steiner triple systems with chromatic index at least (v + 3)=2 and that some of these systems are cyclic.
KW - Block coloring
KW - Chromatic index
KW - Parallel class
KW - Steiner triple system
UR - http://www.scopus.com/inward/record.url?scp=85040375717&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85040375717&partnerID=8YFLogxK
U2 - 10.1137/17M1114338
DO - 10.1137/17M1114338
M3 - Article
VL - 31
SP - 2603
EP - 2611
JO - SIAM Journal on Discrete Mathematics
JF - SIAM Journal on Discrete Mathematics
SN - 0895-4801
IS - 4
ER -